2.23 Example 23 \(\left ( y^{\prime }\right ) ^{2}\left ( 2-3y\right ) ^{2}-4\left ( 1-y\right ) =0\)
\begin{align*} \left ( y^{\prime }\right ) ^{2}\left ( 2-3y\right ) ^{2}-4\left ( 1-y\right ) & =0\\ p^{2}\left ( 2-3y\right ) ^{2}-4\left ( 1-y\right ) & =0 \end{align*}
Since quadratic in \(p\) then
\begin{align*} b^{2}-4ac & =0\\ 0-4\left ( 2-3y\right ) ^{2}\left ( -4\left ( 1-y\right ) \right ) & =0\\ \left ( 2-3y\right ) ^{2}\left ( 1-y\right ) & =0 \end{align*}
Comparing to \(ET^{2}C=0\) shows that \(y=\frac {2}{3}\) is Tac locus and \(y=1\) is \(E\) since it verifies the ode (\(C\) will not verify the
ode).
The following plot shows the singular solution as the envelope of the family of general
solution plotted using different values of \(c.\)